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Journal of Quantitative Economics (Forthcoming)

Is the Indian Capital Market Integrated with Major Developed Economies? Evidence

Using Long Memory Approach

Aviral Kumar Tiwari

Research scholar and Faculty of Applied Economics, Faculty of Management, ICFAI University,

Tripura, Kamalghat, Sadar, West Tripura, 799210, Room No.405, India, Email:

aviral.eco@gmail.com

Amrit Chaudhari (Deceased)

Faculty of Economics, The ICFAI University Tripura, Kamalghat, Sadar, West Tripura, India -

799210. Email: amritchaudhari@gmail.com.

Abstract

The present study attempts to examine the existence of long run diversification benefits of

investment allocation in Indian Stock Market with that of the Developed World by using daily

data of closing stock prices for the period 1997 to 2011 and applying a modified GPH test, along

with a battery of other tests such as KPSS, GPH and the Rescaled Range. The study revealed

conflicting results: the powerful GPH test indicated long memory, while the earlier generation of

tests pointed towards its absence. The results indicate the existence of fractional cointegration of

Indian Stock Market with the Developed world and hence absence of diversification benefits for

the selected markets in the long run.

Keywords: Long Memory, Fractional Co-integration, Modified GPH, KPSS, RS.

JEL Classification: G15.

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1. Introduction

The Stock Market is an important part of a financial system that meets its long-term fund

requirements. In doing so, it also acts as a barometer for the economy. Movements in stock

markets are indicative of the domestic economic conditions as well as the level of confidence

that investors – both domestic and foreign – place in an economy. An important phenomenon

witnessed during the decades after liberalization of the Indian economy in 1991 was the dramatic

increase in its international capital inflows. The Indian economy saw substantial Foreign Direct

Investment (FDI) inflows and portfolio investment being made by Foreign Indirect Investors

(FII). India was termed as a promising emerging market, given the high levels of growth taking

place during the past two decades. This in turn points towards a greater integration of the Indian

stock market with that of the developed world which in turn would have effects on the exchange

rate, trade and the monetary and the fiscal policies of the country. Thus, it becomes necessary to

understand the linkages between the Indian stock market and that of the developed world. With

this background, our study will look into the cointegration between the Indian stock market with

the developed world.

The Efficient Markets Hypothesis (EMH) by Fama (1965) holds that an efficient market

is where security prices adjust rapidly to new information and hence the current prices of

securities would reflect all available information about a security. Thus, while a market can over-

adjust or under-adjust, it would not be biased and an investor would not be able to predict the

occurrences over a period of time. Thus, since market prices would reflect all available

information, the differences between prices at two different points in time cannot be predicted

i.e., a random event. However, practitioners are skeptical about the concept and believe that the

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Indian markets are increasingly becoming integrated with the global economy. However, there

are only a few studies that examine Long memory effects for the same. The paper revisits the

question of whether or not actual stock market prices exhibit long-range dependence in the

context of India and with that of the stock markets of major developed markets, during the post-

liberalization phase of the Indian economy by using Long memory approach.

A number of important reasons for the investors, investment fund managers, policy

makers and academicians make study worthy by understanding of the international linkages of

stock market. This study might be helpful in understanding the benefits that arises from the

international portfolio diversification, stock market efficiency, capital budgeting, formulation of

macro-economic policy and financial regulation. First, the interdependence of stock markets is

helpful for the international investors in understanding the global stock market environment and

to anticipate the risk and rewards due to international diversification. Greater integration among

the world stock markets reduces the potential gain arises due to international diversification.

Second, evidence of high interdependence among the world stock markets might lead us to reject

the efficient markets hypothesis. Third, to study the world stock market integration is important

from the macroeconomic and monetary policies perspective also as stock market co-movements

might lead to market contagion because investors incorporate into their trading decisions

information about price changes in other markets. Fourth, examining the world stock market

integration has fundamental implication in international capital budgeting. For example, in case

of segmented capital markets, the cost of capital for a project will depend upon the source of

finance, whereas this cost will be irrespective of the location where it is raised in an integrated

world market. Last but not least, increasing integration of stock markets of the world can be

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related with numerous benefits such as the development of financial markets and securities

market design, and effective price discovery, higher savings, investment and economic progress.

India has engaged in various bilateral and/or multilateral trade and economic cooperation

agreements with several countries and regional groups across Asia and Europe. Even if, the

Indian economy is closely associated with the rest of world through international trade and

foreign direct investment, the extent of global integration of the Indian stock market is not clear.

The recent crash in the BSE Sensex triggered by US financial crisis is simply an indicator of the

impact of international contagion. It has created substantial interest in studying the

interdependence and dynamic linkages among national stock markets. The main objective of this

paper is to probes the existence of Long memory for the Indian Stock Market, with the

developed markets in Germany, Japan, Switzerland, United Kingdom, and United States. It

considers daily data of closing stock prices from these countries for the time period 1997 to

2011. It uses techniques which are considered robust for testing the long memory component of

markets: the modified GPH test, along with a battery of other tests such as KPSS, GPH and the

Rescaled Range. The study revealed that the data is log non-normal. More importantly, the

results were split: the powerful GPH test indicated long memory, while the earlier generation of

tests point towards its absence. The results indicated the existence of fractional integration of

Indian Stock Market with those of the Developed world. This is suggestive of increasing

integration of the Indian Markets with the developed markets. This also implies an absence of

diversification benefits for these selected markets for the long run.

The plan for the rest of the paper is as follows: Section 2 presents a brief literature review

on stock market cointegration and long memory, Section 3 discusses the data used and the

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methodology employed for analysis and Section 4 discusses the results while Section 5

concludes.

2. Literature Review

As highlighted in the introduction, the interrelationship between various stock markets has

important microeconomic and macroeconomic implications. The various stock markets in

developing countries are having greater amounts of portfolio investment from mature markets as

they are remunerative. Along with globalization, the other contributory factors were the

loosening of financial regulations, introduction of innovative financial products, advancement in

trading systems and improvements in communications networks (Wong et al., 2005).

Following the seminal work of Grubel (1968), there has been substantial interest in

studying the relationships among the financial markets. Early work by Ripley (1973), Lessard

(1976), and Hilliard (1979) found low correlations between national stock markets.

The October 1987 crash focused attention on the connections between various national

equity markets. Eun and Shim (1989) found evidence of co-movements between the US stock

markets and the other national stock markets by using vector autoregression models. Lee and

Kim (1994) analyzed the October 1987 crash and found that national stock markets had become

interrelated after the crash and that the movements among national stock markets were stronger

when the US stock market were more volatile. Jeon and Furstenberg (1990) found that the degree

of international co-movement in stock price indices saw a significant increase after the 1987

crash.

However, there are some studies indicating an absence of integration among the stock

markets. Koop (1994) found no co-movement in stock prices by using Bayesian methods.

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Similarly, Corhay et al. (1995) did not find any evidence of a single stochastic trend for

developed markets. Forbes and Rigobon (2002) show that currency crisis did not increase

integration among stock markets.

There have only been a few studies to look into the co-movement of the Indian stock

market along with the world markets. Sharma and Kennedy (1977) in their study found no

evidence of a cyclical component or periodicity of the Indian stock market along with that of the

US and UK markets. Rao and Naik (1990) found a poor relationship of the Indian market with

the international markets while using Cross-Spectral analysis. However, the Indian stock market

has become much more open to the world since liberalization and there could be a change in the

relationship. Wong et al. (2005) examined the nature of the co-movement between Indian stock

market and main indices to find that the Indian stock market is statistically significantly

cointegrated with the stock markets in the US, UK and Japan by using OLS estimation. They

examined the long-run dynamics of these four markets by using fractionally integrating

technique and found that the Indian stock index forms fractionally cointegrated relationship in

the long run. They emphasize that since only the Johansen Multivariate model can generate the

stationary error term, it shows the superiority of the Johansen method over others. Overall, they

conclude that the Indian financial liberalization opened up the Indian stock markets to the outside

world and its markets are being influenced by other markets. They believe that this reflects the

heavy controls on the Indian economy through the seventies and the liberalization measures

being initiated only in the late eighties. However, substantial changes have taken place since the

time and Indian economy continues to be on a high growth path. Thus, there is a need to revisit

the issue for evidence of Long memory.

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Jagric et al. (2005) found mixed evidence of long memory presence in the stock indices

of six Central and Eastern European (CEE) countries for the period between 1991-2004: strong

long-range dependence was identified in the returns of the Czech, Hungarian, Russian and

Slovenian stock markets, whereas weak or no long-range dependence in the returns of the

Slovakian and Polish stock markets. Assaf (2006) investigated the long memory properties of

stock market returns and volatility for countries of MENA region (that includes Egypt, Jordan,

Morocco, and Turkey) and used fractional integration (non-parametric) procedures proposed by

Lo (1991) and Giraitis et al. (2003). He finds evidence of long memory in the stock returns for

Egypt and Morocco. For Jordan and Turkey, he finds evidence of anti-persistence. He also finds

evidence of long memory in the volatility series for the entire sample countries examined. Jagric

et al. (2006) investigated long memory in stock returns of ten CEE countries. Their results

suggested strong long memory presence in eight stock markets, among them in the Croatian and

Hungarian stock markets. Gurgul and Wójtowicz (2006) investigated the long-memory

properties of trading volume and the volatility of stock returns (given by absolute returns and

alternatively by square returns) for the daily stock data of German companies in the DAX

segment of the German Stock Exchange. Authors, performed analysis for daily data covering

period from January 1994 to November 2005 and in three sub-periods: January 1994 to

December 1997, January 1998 to December 2001, and January 2002 to November 2005. The

study found that for the equities listed in the DAX index the log-volume and returns volatility

exhibit long memory. Moreover these two series have the same long-memory parameters for

most of the equities. This common long memory of both series is especially strongly pronounced

in the latest data. On the other hand, there is no evidence that log-volume and volatility share the

same long memory component. Caporale and Gil-Alana (2007) examined the long-run dynamics

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and the cyclical structure of various series (such as inflation, real risk-free rate, real stock returns,

equity premium, and price/dividend ratio) related to the US stock market using fractional

integration for the period from 1871 to 1993. They found that the estimated order of integration

varies considerably, but nonstationarity is found only for the price/dividend ratio, when focusing

exclusively on the long-run or zero frequency. When the authors taken into account the cyclical

component, they found that series are stationary and exhibit long memory with respect to both

components in almost all cases with the exception to the price/dividend ratio. Supportive

evidence of long memory is provided by Ozdemir (2007) and Chan and Feng (2008) for the DJI,

the S&P500, the FTSE, DAX and NIKKEI (for different time periods), Bilel and Nadhem (2009)

for G7 countries stock indices for high frequency data between 2003-2004 and Mariani et al.

(2010) for international stock indices. Kasman et al. (2009a) investigated the fractal structure of

the CEE stock markets. The authors found the existence of long memory in five of eight studied

markets. Furthermore, Kasman et al. (2009b), investigating four CEE stock markets, found

significant long memory in the return series of the Slovak Republic, weak evidence of long

memory for Hungary and the Czech Republic, and no evidence for Poland.

Lim (2009) tested the efficient market hypothesis for Middle East and African countries

using non-linear models. He finds that the returns for emerging stock markets for both Middle

East and Africa contain predictable components, and therefore are inefficient. Alagidede and

Panagiotidis (2009) investigated the random walk behavior of stock market returns for Egypt,

Kenya, Morocco, Nigeria, South Africa, Tunisia and Zimbabwe. They reject the null hypothesis

that the stock market returns for the sample countries are random walk processes. However,

using smooth transition and conditional volatility models they find evidence of volatility

clustering, leptokurtosis and leverage effect in the African data. Karanasos and Kartsaklas (2009)

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investigated the issue of temporal ordering of the range-based volatility and turnover volume in

the Korean market for the period 1995–2005 and also for the period before financial crisis and

after financial crisis. The authors examined the dynamics of the two variables and their

respective uncertainties using a bivariate dual long-memory model. The authors found that the

apparent long-memory in the variables was quite resistant to the presence of breaks and when

they take into account structural breaks the order of integration of the conditional variance series

decreases considerably. Further, the authors showed that the impact of foreign volume on

volatility was negative in the pre-crisis period but turns to positive after the crisis. The authors

found that before the crisis there was no causal effect for domestic volume on volatility whereas

in the post-crisis period total and domestic volumes affect volatility positively.

Tan et al. (2010) examined the efficient market hypothesis on the Malaysian stock market

under both bullish and bearish periods covering from 1985:01 to 2009:12. They first identified

the bullish and bearish periods by using the Bry and Boschan (1971) algorithm, following by

Geweke and Porter-Hudak’s (1983) test for the diagnosis of market efficiency. Their study

provided the evidence of persistent long memory under the earlier periods of the study,

suggesting the possibility to predict stock prices especially before the 1997 financial crisis. Tan

et al. (2011) examined the predictability in stock return in developed and emerging markets by

testing long memory in stock returns using wavelet approach. Wavelet-based maximum

likelihood estimator of the fractional integration estimator is superior to the conventional Hurst

exponent and Geweke and Porter-Hudak estimator in terms of asymptotic properties and mean

squared error. The authors used 4-year moving windows to estimate the fractional integration

parameter. They suggested that stock return may not be predictable in developed countries of the

Asia-Pacific region. However, predictability of stock return in some developing countries in this

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region such as Indonesia, Malaysia and Philippines may not be ruled out. Stock return in the

Thailand stock market appears to be not predictable after the political crisis in 2008. Festić et al.

(2012) analyzed and compared the fractal structure of the Croatian and Hungarian stock market

returns. The presence of long memory components in asset returns provides evidence against the

weak-form of stock market efficiency. The study provided the evidence that the wavelet ordinary

least squares (WOLS) estimator may lead to different conclusions regarding long memory

presence in the stock returns from the KPSS unit root tests and/or Lo’s R/S test. Furthermore, it

proved that the fractal structure of individual stock returns may be masked in aggregated stock

market returns (i.e. in returns of stock index). The main finding of the study was that the

Croatian stock index, Crobex, and individual stocks in this index exhibited long memory. Long

memory is identified for some stocks in the Hungarian stock market as well, but not for the stock

market index BUX. Based on the results of the long memory tests, the study concluded that

while the Hungarian stock market is weak form efficient, the Croatian stock market is not.

The literature review above clearly shows that there is conflicting evidence on the issue of

international stock market linkages, depending on the methodology, data, and time period

studied. Unfortunately, little empirical evidence exists concerning linkages between Indian stock

market and other markets of the world and that uses long memory tests. In this regard India has

much less exposure in the stock market integration literature until recently. Given India’s fast-

growing economic influence, research on the Indian stock market still seems to be inadequate

and needs further investigation using approach of long memory tests. The present study extends

the existing stock market integration literature in the following ways. First, to provide further

evidence, we examine the stock price linkages i.e., interdependence between the stock market of

India and that of the Germany, Japan, Switzerland, United Kingdom and the U.S. using daily

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stock closing price indices data covering the period July 1, 1997 to April 29, 2011. Second, this

research is based on the long memory approach and a battery of tests is utilized to examines

interdependence of Indian stock market to the developed world and provide robust evidence.

And lastly, from the implication and policy point of view, the results from this research provide

implications regarding international diversification and market efficiency that are important for

investors and fund managers who are interested in investing in these markets and from the

macroeconomic and monetary policies perspective as stock market co-movements might lead to

market contagion because investors incorporate into their trading decisions information about

price changes in other markets.

3. Data and Methodology

The paper has used daily closing stock prices data for the period July 1, 1997 to April 29, 2011.

We have used the following proxies for the markets: Sensex for India, DAX for Germany,

Nikkei for Japan, SSMI for Switzerland, FTSE for United Kingdom and the NYSE Composite

Index for the U.S. All data was taken from the Yahoo Finance (www.finance.yahoo.com). The

total number of observations was 3371 after removing the data for the days in which trading have

not occurred at least in one stock market. Our choice of country’s stock markets is guided by the

consideration that India has significant trade and financial relations with these countries or they

are big players in the international stock market and thereby able to transmit shocks to other

stock markets if they experience it and also as Tiwari and Islam (2012) provided a

comprehensive review in their study which show that most of the studies those have conducted

test of cointegration to analyses the stock market integration have mostly tested with those group

of countries we analyzed in the present work. The period of the data is dependent upon the

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availability of the data from the Yahoo Finance. Data is converted into natural logarithms before

any computation is performed. The features of the data are described in Table 1 and trend plot of

the variables is presented in Figure 1.

Table 1: Summary Statistics of Daily Closing Prices of Stock Market Indices (values in

natural log form)

India Germany Japan Switzerland United Kingdom USA Mean 8.832401 8.557555 9.464249 8.792090 8.576939 8.817620 Median 8.642667 8.583707 9.471687 8.794628 8.599750 8.808061 Maximum 9.952514 9.000322 9.944304 9.162357 8.843644 9.241026 Minimum 7.863313 7.697557 8.861489 8.209417 8.097731 8.349085 Std. Dev. 0.656400 0.259106 0.259367 0.179181 0.157085 0.188554 Skewness 0.293700 -0.500600 -0.066301 -0.200984 -0.537008 0.170084 Kurtosis 1.534729 2.729016 1.801186 2.574537 2.311651 2.514957 Jarque-Bera Probability

350.0305 (0.000000)

151.1096 (0.000000)

204.3302 (0.000000)

48.12058 (0.000000)

228.5728 (0.000000)

49.29824 (0.000000)

It is evident from Table 1 that the sample means of level data set is positive for all the countries

The measure of skewness indicates that in the log level form stock indices of Germany, Japan,

Switzerland and U.K. are negatively skewed whereas for India and USA is positively skewed.

Data series in all log level series have demonstrated less kurtosis which indicates that

distributions of stock prices of countries analysed are Platykurtic relative to a normal

distribution. The Jarque-Bera normality test rejects normality of all series at any level of

statistical significance indicating that stock prices are not log-normally distributed.

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Figure 1: Time Series Plot of Daily Closing Prices of Stock Market Indices (July 1, 1997 to April 29, 2011) (values in natural log form)

7.6

8.0

8.4

8.8

9.2

9.6

10.0

500 1000 1500 2000 2500 3000

INDIA GERMANYJAPAN SWITZERLANDUNITEDKINGDOM USA

From the graphical exposition (depicted in Figure 1), which depicts the movements of

different stock market indices during the period July 1, 1997 to April 29, 2011, it is evident that

there is quite close association between trends in Indian stock market and the stock markets of

the developed World which has experienced the various phases of ups and downs during the

study period.

Long memory describes the correlation of a series after long lags, which shows temporal

dependence over distant observations. Long memory would arise from non-linear dependence in

the first movement of the distribution and denotes existence of a predictable component in the

time series distribution. The existence of long memory in stock prices gives evidence against the

weak form of market efficiency. Long Memory process or long range dependence holds that past

events that occurred a long time ago can have an influence on events today. It holds that

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correlations between observations far apart in time decay to zero at a rate that is much slower

than the one expected from independent data or data that is modeled according to the classical

ARMA or Markov frameworks. In other words, the dependence between events far apart

diminishes slowly with an increasing distance. Beran (1994, p.6) remarks “a stationary process

with slowly decaying correlations which is un-summable is, therefore, called a stationary process

with long memory or long-range dependence or strong dependence (in contrast to processes with

summable correlations which are also called processes with short memory or short-range

dependence or weak dependence)”. The long memory process can be seen existing half-way

between the I(0) and I(1), implying different long run predictions than conventional approaches.

We discuss the techniques of the KPSS test, Fractional Cointegration Analysis, Mean reversion,

Martingale Variance Ratio Test, Long memory, Mean Reversion and Rescaled Range Analysis

below.

3.1 KPSS Test

Lee and Schmidt (1996)1 propose the test of Kwiatkowski et al. (1992), hereafter KPSS, as a test

for the null of stationarity against the alternative hypothesis of fractional integration. The KPSS

test involves regressing tre against a constantµ , and a trendτ . The test is based upon

)(/22 qSTTt∑−= σητ (1)

1 The Kwiatkowski, Phillips, Schmidt, and Shin (KPSS, 1992) test for stationarity of a time series differs from those “unit root” tests in common use (such as ADF, PP, and DF-GLS) by having a null hypothesis of stationarity. The test may be conducted under the null of either trend stationarity or level stationarity. Inference from this test is complementary to that derived from those based on the Dickey-Fuller distribution (such as ADF, PP, and DF-GLS). The KPSS test is often used in conjunction with those tests to investigate the possibility that a series is fractionally integrated (that is, neither I(1) nor I(0)); see Lee and Schmidt (1996). As such, it is complementary to GPH, and modified rescaled range statistic.

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where 2tS represents the square of the partial sum of the residuals and )(qTσ represents an

estimate of the long run variance of the residuals. Lee and Schmidt (1996) present Monte Carlo

evidence that the KPSS test has power properties similar to the modified rescaled range statistic

of Lo (1991).

3.2 Fractional Integration Analysis

The process was developed independently by Granger (1980), Granger and Joyeux (1980) and

Hosking (1981). It is also termed as the ARFIMA(p, d, q) model. Here, d denotes the fractional

differencing parameter, and is written as

Φ (L)(1− L )d (yt − µ) = Θ (L )ε, εt ~ i.i.d.(0,σ2ε) (2)

where L is the backward-shift operator. The parameter d assumes any real value. For d≠0 the

data series is I(d). The stochastic process yt, is both stationary and invertible if all roots of Φ(L)

and Θ(L) lie outside the unit circle and | d |< 0.5 . The process is nonstationary for d ≥ 0.5 as it

possessed infinite variance. For d ∈(0,0.5), the ARFIMA process exhibits long memory or long-

range positive dependence. The process exhibits intermediate memory or long-range negative

dependence for d ∈(− 0.5,0) and short memory for d = 0.

While the usual notion of integration has the strict I(0) and I(1) distinction, fractional

integration allows the variables to be fractionally integrated.2 A flexible and parsimonious way to

model both the short-term and the long-term behavior of a time series ),...,( 1 Tt ZZZ = and its first

2 In this study we computed a modified form of the Geweke and Porter-Hudak (GPH, 1983) estimate of the long memory (fractional integration) parameter, d, of a time series, proposed by Phillips (1999a, 1999b). This is because if a series exhibits long memory, then distinguishing unit-root behavior from fractional integration may be problematic, given that the GPH estimator is inconsistent against d>1 alternatives. This weakness of the GPH estimator is solved by Phillips’ (1999a, 1999b) Modified Log Periodogram Regression estimator, in which the dependent variable is modified to reflect the distribution of d under the null hypothesis that d=1. The estimator gives rise to a test statistic for d=1 which is a standard normal variate under the null. Phillips suggests (1999a, 1999b) that deterministic trends should be removed from the series before application of the estimator.

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difference, tX , denoted tt ZLX )1( −= , where L is the lag (or backward-shift) operator by way

of an autoregressive fractionally-integrative moving-average (ARFIMA) process of order

( qdp ,, ) becomes:

),,0(~)()1)(()1)(( 2,

' σiidLXLLZLL tttd

td ∈∈Θ=−Φ=−Φ (3)

where d is a fractional difference operator and possibly a non-integer; and )(LΦ and )(LΘ are

the autoregressive and moving average lag polynomials of orders p and q , respectively, [i.e.,

dqq

pp LandddLLLLLL )1(,'1,...1)(,...1)( 11 −+=+++=Θ−−−=Φ φφφφ is the fractional

differencing operator defined by ∑∞

=−Γ+Γ−Γ=−

0)()1(/)()1(

k

kd dkLdkL with

)(⋅Γ denoting the gamma or generalized factorial function] with roots lying outside the unit

circle. An integer value of 0=d yields a standard ARMA process (i.e., the process exhibits

short memory for 0=d , whereas 1=d gives rise to the unit-root nonstationary process. For

)5.0,0(∈d and 0≠d , the ARFIMA process is said to exhibit long memory and the

autocorrelations of the ARFIMA process decay hyperbolically to zero as .∞→k For

)0,5.0(−∈d , the ARFIMA process is said to exhibit intermediate memory (see Granger and

Joyeux, 1980). Although for )1,5.0(∈d the process implies mean reversion, however, it is not a

covariance stationary process. Finally 1>d implies that the process is not mean reverting. The

fractional integration test due to Geweke and Porter-Hudak (1983) is used to detect the nature of

stationarity. Geweke and Porter-Hudak (1983) demonstrate that the fractional differencing

parameter d can be estimated consistently from the least squares regression at frequencies near

zero:

,))2/(sin4ln())(ln( 2jjj vI ++= ωβαω Jj ,...,1= (4)

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where α is the constant term, )1,...,1(/2 −== TjTjj πω TTTJ ,<<= η is the number of

observations, and )( jI ω is the periodogram of time series tX at frequency jω . With a proper

choice of J, the negative of the OLS estimate of β coefficient gives a consistent and

asymptotically normal estimate of the order of integrationd .

Moreover, this is true regardless the orders and the estimates of the parameters of the

ARMA process. While 5.0=η is suggested in the empirical analysis, we set η also equal to

0.475 and 0.525 for checking the sensitivity of the results to the selection of η .3 GPH test is

conducted on the log transformed series of the data. As d of the level series equals ,'1 d+ a

value of 'd equal to zero corresponds to a unit root in tZ . Thus, the GPH test is used as a unit

root test as we apply it to the first differences of the relevant series. The unit root null hypothesis

1=d )0'( =d d can be tested against the one-sided, long-memory, fractionally-integrated

alternative 1<d )0'( <d . The rejection of the unit-root null hypothesis would suggest the

existence of fractional integration.

An alternative semi-parametric approach is that of Robinson (1992), who considers the

discretely averaged Periodogram

∫= λλθ dfF )()( (5)

Robinson estimates the Hurst coefficient, or equivalently the degree of fractional

integration, d, as

3 Cheung and Lai (1993) conducted a Monte Carlo experiment and found better performance for η = 0.55, 0.575,

and 0.6. In another investigation, Cheung (1993) used η = 0.5 (which is commonly used to test for fractional

integration), and also reports results for η = 0.45 and 0.55 to check the sensitivity of the estimates. Overall, it may

be inferred that, irrespective of sample size, a value of η , between 0.5 and 0.6 appears to be the ideal choice.

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)(/)(log{)}log(21{ 1m

wq wFqFqH m−−= (6)

where )1,0(∈q , )(ˆmwF is the average of periodogram with )( jwI , with Tjw j /2π= for j = 1, …,

m < T/2, and n is the sample size. Specifically, ∑=

−=]2/[

1

1 )(2)(ˆπ

πwT

jjm wInwF where [ . ] means

the integer part. Robinson (1994) shows that the measure d is consistent as T gets large. We use

q = 0.5 in this study. As with the rescaled range and GPH approaches, there is a substantial

amount of evidence documenting the poor performance of the Robinson’s semi-parametric

estimator in terms of bias (see Baillie, 1996).

3.3 Mean Reversion, Martingale and Variance Ratio Test

To examine further, whether the cointegrating series of stock markets follows a martingale, we

employ a joint variance ratio test based on Lo and MacKinlay (1988). The Lo and MacKinlay

(1988, 1989) overlapping variance ratio test, examines the predictability of time series data by

comparing variances of differences of the data (returns) calculated over different intervals. If we

assume the data follow a random walk, the variance of a q -period difference should be q times

the variance of the one-period difference. Evaluating the empirical evidence for or against this

restriction is the basis of the variance ratio test. Diebold (1989) show that the joint variance ratio

test used in this study has good power against fractionally integrated alternatives.

The variance ratio test makes use of the fact that the variance of the increments in a

martingale is linear in the sampling interval.4 That is, if the time series tZ follows a martingale

4 Lo and MacKinlay (1988) formulate two test statistics for the random walk properties that are applicable under

different sets of null hypothesis assumptions about t∈ . First, Lo and MacKinlay make the strong assumption that

the t∈ are i.i.d. Gaussian with variance 2σ (though the normality assumption is not strictly necessary). Lo and

18

process, the variance of its thq − differences would be q times the variance of its first

differences. Using a daily interval as the base in our study, the variance ratio at lag q , VR(q), is

defined as

,/)( 21

2 σσ qqVR q= (7)

where 2qσ is an unbiased estimator of the variance of the thq − difference of tZ and 2

1σ is an

unbiased estimator of the variance of the first difference of tZ . Thus, under the martingale

hypothesis, VR(q) equals one for any q chosen. The variance ratio estimate is closely related to

the serial correlation of the series as the variance-ratio estimate for a given q is approximately a

linear combination of the first 1−q autocorrelation coefficients (Cochrane, 1988). Accordingly,

rejecting the null hypothesis of martingale with a variance ratio of less (greater) than one implies

negative (positive) serial correlations, or mean-reversion (mean-aversion). Thus, the VR statistic

provides an intuitively appealing way to search for mean reversion, an interesting deviation from

the martingale hypothesis. The hypothesis is tested under the asymptotic distributions of both

homoskedasticity-and heteroskedasticity-robust variance ratio estimators developed by Lo and

MacKinlay (1988). Nevertheless, Lo and MacKinlay’s (1988) asymptotic distribution might not

be an accurate approximation when q is large and sample size is small (Richardson and Stock,

1990). We thereby provide the p value of the VR statistics in testing the null by using the

bootstrap method.

MacKinlay term this the homoskedastic random walk hypothesis, though others refer to this as the i.i.d. null. Alternately, Lo and MacKinlay outline a heteroskedastic random walk hypothesis where they weaken the i.i.d. assumption and allow for fairly general forms of conditional heteroskedasticity and dependence. This hypothesis is

sometimes termed the martingale null, since it offers a set of sufficient (but not necessary), conditions for t∈ to be a

martingale difference sequence (m.d.s.).

19

Furthermore, in earlier works (i.e., Liu and He, 1991), the martingale hypothesis is

considered rejected when at least some of the VR statistics provide evidence against it.

Richardson (1993) notes that the failure of including a joint test that combines all of the

information from several VR statistics would tend to yield stronger results. To provide a joint test

that takes account of the correlations between VR statistics at various horizons, we consider the

two approaches one by Chow and Denning (1993) and second one by Richardson and Smith

(1991). This is because the variance ratio restriction holds for every difference 1>q , it is

common to evaluate the statistic at several selected values of q . To control the size of the joint

test, Chow and Denning (1993) propose a (conservative) test statistic that examines the

maximum absolute value of a set of multiple variance ratio statistics. The p-value for the Chow-

Denning statistic using m variance ratio statistics is bounded from above by the probability for

the Studentized Maximum Modulus (SMM) distribution with parameter m and T degrees-of-

freedom. Following Chow and Denning, we approximate this bound using the asymptotic

)( ∞=T SMM distribution. A second approach is available for variance ratio tests of the ... dii

null. Under this set of assumptions, we may form the joint covariance matrix of the variance ratio

test statistics as in Richardson and Smith (1991), and compute the standard Wald statistic for the

joint hypothesis that all m variance ratio statistics equal 1. Under the null, the Wald statistic is

asymptotic Chi-square with m degrees-of-freedom.

However, there is possibility that the empirical distributions of VR statistics might have a

large degree of positive skewness, suggesting that inference based on the 2χ distribution will be

misleading. Accordingly, we calculate the Wald statistic for each bootstrapped VR estimator

20

vector and use the bootstrapped distribution of Wald statistics for hypothesis testing by using

Wild Bootstrap of Kim (2006).5

3.4 Long memory, mean reversion and rescaled range analysis

To further examine the characteristics of long-memory and mean-reversion in the cointegrating

series of stock markets, we employ the rescaled range (R/S) test. The R/S statistic is formed by

measuring the range between the maximum and minimum distances that the cumulative sum of a

stochastic random variable has deviated from its mean and then dividing this by its standard

deviation. An unusually small (large) R/S statistic signifies mean-reversion (mean-aversion).

Mandelbrot (1972) demonstrates that the R/S statistic can uncover not only periodic dependence

but also non periodic cycles. He further shows that the R/S statistic is a more general test of

long-memory in time series than examining either autocorrelations (i.e., the variance ratio test) or

spectral densities.

Lo (1991) points out that the original version of the R/S analysis (which may be termed

as classical R/S test) has limitations in that it cannot distinguish between short and long-term

dependence, nor is it robust to heteroskedasticity. Lo (1991) developed a test for short memory

versus long memory based on a simple modification of the rescaled range or RS statistic

introduced by Hurst (1951). The modified RS statistic is

5 Kim (2006) offers a wild bootstrap approach to improving the small sample properties of variance ratio tests. The approach involves computing the individual (Lo and MacKinlay) and joint (Chow and Denning, Wald) variance ratio test statistics on samples of observations formed by weighting the original data by mean 0 and variance 1 random variables, and using the results to form bootstrap distributions of the test statistics. The bootstrap p-values are computed directly from the fraction of replications falling outside the bounds defined by the estimated statistic. E-views offer three distributions for constructing wild bootstrap weights: the two-point, the Rademacher, and the normal. Kim’s simulations indicate that the test results are generally insensitive to the choice of wild bootstrap distribution.

21

,)()(ˆ1

1111

∑ −−∑ −=

=≤≤=≤≤

k

tt

Tk

k

tt

Tkn XXMinXXMaxQ

ω (8)

Under the null hypothesis, Lo (1991) showed that

)(min)(max 0

1

0

1tWtWU

n

QRS

nknkRS

n

≤≤≤≤−=⇒= .

The distribution function of the random variable RSU was derived by Feller (1951) and has the

formula

∑∞

=

−−+=1

222 22

)41(21)(k

xkU exkxF

RS .

This distribution function is used to calculate the critical values of the modified RS test. Critical

values at various significance levels are available in Table 2 in Lo (1991). Further, for estimation

in our analysis we considered two lag truncation procedures: a lag truncation equals to zero and

the Andrews’ formula.6

6 In practice, however, a bandwidth value q has to be selected in the construction of the tests. Consequently, the finite sample performance of these tests depends on the choice of the bandwidth. The most popular bandwidth

choice is probably the data-dependent automatic bandwidth )12/(1),(ˆ += p

k nkfq δµ where kµ is a

constant associated with the kernel function k, ),( kfδ is a function of the unknown spectral density and is

estimated using a plug-in method, and p is the characteristic exponent of k. This bandwidth choice has been studied by Andrews (1991) in the estimation of a covariance matrix for stationary time series and is now widely used in econometrics applications. It has the advantage that it partially adapts to the serial correlation in the underlying time

series through the data-dependent component ),(ˆ kfδ . Lima and Xiao (2010) find that the problem of choosing the

optimal q is not solved yet, but providing a bandwidth procedure that is robust against both the null and alternative models may help practitioners in their investigation on the presence of long memory in financial time series.

22

4. Results

To achieve our objective i.e., to test for fractional cointegration, if it exists, we first run simple

OLS regression using Indian stock market index as dependent variable and other stock market of

the developed world as independent variables, sequentially, (both stock indices are measured in

natural log form) and then saved residuals as error-correction term. In the second step we applied

a modified GPH test, along with a battery of other tests such, GPH and the Rescaled Range to the

first differences of the error correction terms test to test for the evidence of cointegration.

Evidence of fractional cointegration will be found if null of unit root is rejected for the first

differenced OLS residual series. We presented in Table 2 the summary statistics for the OLS

residuals estimated from simple OLS regression analysis for the period under consideration in

this study.7

Table 2: Summary Statistics and KPSS Unit Root Results for the OLS Regression

Residuals

Countries Mean Std. Dev. Skewness Kurtosis

Jarque-Bera

(p-values)

KPSS

µη τη

India-Germany -4.59E-14 0.546 -0.3238 1.796716 262.277 (0.00) 36.7(6) 2.41(6)

India-Japan 7.39E-15 0.650 0.26161 1.57615 323.211 (0.00) 33.4(6) 4.96(6)

India-Switzerland 2.39E-14 0.621 0.22040 1.800454 229.399 (0.00) 39.6(6) 4.34(6)

India-United kingdom 1.07E-14 0.635 0.18151 1.544387 316.115 (0.00) 40.7(6) 4.63(6)

India-USA -2.78E-14 0.463 0.80571 3.026776 364.833 (0.00) 26.1((6) 4.24(6)

Note: ***, ** and * indicate significance at the 0.01, 0.05 and 0.1 level, respectively. In case of KPSS test the lag length selection is based on the truncation for the Bartlett Kernel, as suggested by the Newey-West test (1987). Critical values for KPSS test at 1%, 5% and 10% level of significance 0.739, 0.463 and 0.347 respectively.

7 Diebold and Rudebusch (1991) demonstrate that the standard unit root tests have low power against fractional alternatives.

23

It appears from Table 2 that OLS residual series are extremely non-normal. India’s stock

market cointegrating series with Germany is negatively skewed, possibly due to the large

downturn associate with this stock market and with the rest of the developed countries it is

positively skewed, possibly due to the large upturn associate with international market. The data

also display a relatively less degree of excess kurtosis except with USA, indicating that OLS

residual series are Platykurtic relative to a normal distribution. Such skewness and kurtosis are

common features in such financial dataset. For all country case OLS residual series fail to satisfy

the null hypothesis of normality of the Jarque-Bera test at the 1% level of the significance.

However, non-stationarity does not preclude the possibility of long-memory in the India’s stock

market cointegrating series with major developed countries.

Table 2 also presents the results of the KPSS (Kwiatkowski et al., 1992). Lee and

Schmidt (1996) argue that KPSS test is consistent against stationary long memory alternatives

and may therefore be used to distinguish between short and long memory processes. The KPSS

test is similar to Lo’s modified rescaled range statistic in both power and construction (see Lee

and Schmidt, 1996 and Baillie, 1996 for further details). However, through KPSS test we find

that India’s stock market cointegrating series with major developed countries have long memory

as in none of the case the stock market cointegrating series with major developed countries

satisfy the null hypothesis of stationarity about a (possibly non-zero) mean of the µη and τη test

at the 1% level of significance. These results suggest that India’s stock market cointegrating

series with major developed countries contain a nonstationary component. In short, the evidence

from the KPSS tests is in favor of the presence of long memory in the all series of India’s stock

market cointegrating series with major developed countries.

24

Baillie (1996) comments that the GPH estimator is potentially robust to non-normality as

our data set also exhibits that data (both raw data as well as residual series) is non-normal. Table

3 displays the results of the GPH and MGPH estimates of d, the degree of fractional integration,

along with F-statistics for the null hypotheses of 1=d . A concern in the application of the GPH

and MGPH estimator is the choice of J (as in most cases, the results vary across the different

values of η ), the number of spectral ordinates from the periodogram of tX , to include in the

estimation of d. Here results are presented for the first differenced OLS residual series for

ηTJ = ; where 6.0,55.0,5.0=η , where T represents the sample size (i.e., number of

observations).

Table 3: Results of the Geweke-Porter-Hudak (GPH) and Modified GPH (MGPH) Test for Fractional Integration and Robinson Estimates of Rescaled Range8

Countries MGPH estimates

Rescaled range

(R/S) analysis of

long memory

η =0.5 η =0.55 η =0.6 Lag=0 Lag=1

d H0: d=0 H0: d=1 D H0: d=0 H0: d=1 d H0: d=0 H0: d=1

India-Germany .415

5.52

(0.00)

-6.94

(0.00) .321

4.78

(0.00)

-9.86

(0.00) .211

3.99

(0.00)

-14.01

(0.00) .998 .998

India-Japan .159

2.13

(0.03)

-9.97

(0.00) .153

2.42

(0.01)

-12.3 1

(0.00) .120

2.19

(0.03)

-15.64

(0.00) 1.59 1.55

India-

Switzerland .334

4.40

(0.00)

-7.90

(0.00) .209

3.39

(0.00)

-11.49

(0.00) .178

3.36

(0.00)

-14.60

(0.00) .981 .977

India-United

kingdom .281

3.07

(0.00)

-8.53

(0.00) .105

1.44

(0.15)

-13.0

(0.00) .054

0.99

(0.32)

-16.81

(0.00) 1.04 1.05

India-USA .318

3.19

(0.00)

-8.09

(0.00) .211

2.47

(0.01)

-11.46

(0.00) .130

1.99

(0.04)

-15.46

(0.00) 1.1 1.1

8 Agiakloglou et al. (1993) suggest that the estimate of the order of fractional integration from the GPH method could be biased for a model with large ARMA parameters. However, if the estimates d remain stable for different η ’s, there is no hint of a bias due to ARMA parameters (Hassler and Wolters, 1995).

25

GPH estimates

India-Germany .043

0.44

(0.65)

0.45

(0.64) .017

0.20

(0.83)

0.22

(0.82)

-

.032

-0.53

(0.59)

-0.54

(0.58)

India-Japan .097

1.25

(0.21)

1.03

(0.30) .105

1.63

(0.10)

1.39

(0.16) .084

1.52

(0.13)

1.40

(0.16)

India-

Switzerland .028

0.46

(0.64)

0.30

(0.76)

-

.036

-0.73

(0.46)

-0.49

(0.62)

-

.007

-0.15

(0.88)

-0.12

(0.90)

India-United

kingdom -.026

-0.32

(0.74)

-0.27

(0.78)

-

.129

-2.03

(0.04)

-1.72

(0.08)

-

.116

-2.35

(0.02)

-1.93

(0.05)

India-USA .029

0.35

(0.72)

0.31

(0.75)

-

.014

-0.20

(0.84)

-0.19

(0.84)

-

.038

-0.65

(0.51)

-0.63

(0.52)

Note: The GPH and MGPH test statistics are F-statistics from the spectral regression. The null hypotheses of 1=d and 0=d are tested

against the alternatives of 1≠d and 0≠d , respectively. Values in the parenthesis are the p-values those are reported below the F-statistics.

Results reported in Table 3 of the GPH and MGPH estimates suggest that the results are

not very much sensitive to the choice of η at 5% level of significance (with only exception for

U.K. for choice of 6.0,55.0=η ). Interestingly, we have contrary results for GPH and MGPH

estimates. The null hypotheses of d=0 and 1=d are rejected in all cases when we applied

MGPH whereas these null hypotheses are not rejected in all cases when we applied GPH

estimates. Worthy to mention that the null hypothesis of d=0 is rejected for GPH estimates and

the null hypothesis of d=0 is not rejected for MGPH estimates only for U.K. for the choice of

6.0,55.0=η . As we know that MGPH is more powerful test statistics, we relied upon the results

of MGPH only to draw conclusions. The rejection of the null hypothesis of d=0 in all cases

imply the presence of unit root in the first differenced OLS residual series i.e., nonstationary of

India’s stock market cointegrating series with major developed countries.

26

However, the rejection of the null hypothesis of 1=d irrespective of the value of η ,

implying that the error-correction in all cases are clearly fractionally integrated (i.e., long

memory), suggesting that the Indian stock market is fractionally cointegrated with the developed

countries. Further, for the cases where the value of )1,5.0(∈d despite the choice of the value of

η implies that though the error-correction term is mean reverting (i.e., cointegrated), however, it

is not a covariance stationary process.

Table 3 also displays the estimates of d calculated using the Gaussian semiparametric

estimator of Robinson (1992). For )5.0,0(∈d (i.e., for 5.00 << d ) the series exhibits long

memory in the cointegration series. The Robinson’s (1992) estimates of Rescaled range (R/S)

analysis of long memory of d are outside the long memory range and close to 1 for all countries

(and for Lag=0 and Lag=1), supporting the findings obtained from GPH test.

5. Conclusions

The study used the techniques of fractional co-integration involving the modified GPH test,

along with a battery of other tests such as KPSS, GPH and the Rescaled Range for India,

Germany, Japan, Switzerland, United Kingdom, and United States for the time period 1997 to

2011. Though the study gave contradictory result, with the earlier tests pointing to an absence of

long memory, the powerful GPH test indicated long memory. The estimation confirms our

hypothesis of the existence of fractional cointegration among the five pairs of stock markets,

namely India-Germany, India-Japan, India-Switzerland, India-U.K. and India-U.S.A.

27

The results indicate that they conclude that the Indian financial liberalization opened up

the Indian stock markets to the outside world and its markets are being influenced by other

markets.

The confirmation of fractional cointegration by this paper should guide investors in their

investment decisions regarding the Indian stock market. The results could be specifically used

for both fundamental and technical analysis of stock markets. Since stock markets are a

barometer for the economy, the results can also help us examine the overall economic situation

and the existence of problems in the financial system.

It should be kept in mind that the use of daily closing stock indices would mean that the

limits for daily price could be affected by the closing price of the previous day. It is plausible

that the data would have hit a circuit breaker and thus be contaminated. Even so, daily price data

is crucial to the understanding of the behavior of the series. It should be pointed out that relations

among stock markets are also affected by macroeconomic variables such as trade and levels of

foreign exchange. Future research should incorporate relevant macroeconomic series which

might provide more insight in to the working of equity market and/or some new tests of

fractional cointegration can be used those incorporates structural breaks in the data set.

Acknowledgement

We are very thankful to an anonymous referee for his/her constructive comments and

suggestions on the earlier version of the manuscript. We strongly feel that this paper would have

remained flawed unless those suggestions were incorporated. We have incorporated all of his/her

comments to the best of our abilities. Error remains, if any, is the authors sole responsibility. The

usual disclaimer applies.

28

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